AMC8 2008

AMC8 2008 · Q20

AMC8 2008 · Q20. It mainly tests Fractions, GCD & LCM.

The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and $\frac{3}{4}$ of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?

尼特金先生班上的学生参加了一次笔迹测试。男生中有 $\frac{2}{3}$ 和女生中有 $\frac{3}{4}$ 通过了测试，并且通过测试的男生和女生人数相等。班上学生的最小可能人数是多少？

(A) 12 12

(B) 17 17

(D) 27 27

(E) 36 36

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Because $\frac{2}{3}$ of the boys passed, the number of boys in the class is a multiple of 3. Because $\frac{3}{4}$ of the girls passed, the number of girls in the class is a multiple of 4. Set up a chart and compare the number of boys who passed with the number of girls who passed to find when they are equal. \[ \begin{array}{|c|c|} \hline \text{Total boys} & \text{Boys passed} \\ \hline 3 & 2 \\ \hline 6 & 4 \\ \hline 9 & 6 \\ \hline \end{array} \qquad \begin{array}{|c|c|} \hline \text{Total girls} & \text{Girls passed} \\ \hline 4 & 3 \\ \hline 8 & 6 \\ \hline \end{array} \] The first time the number of boys who passed equals the number of girls who passed is when they are both 6. The minimum possible number of students is $9+8=17$.

答案（B）：因为男生中有$\frac{2}{3}$通过，所以班里男生人数是3的倍数。因为女生中有$\frac{3}{4}$通过，所以班里女生人数是4的倍数。列一个表，比较通过的男生人数与通过的女生人数，找出它们相等时的情况。 \[ \begin{array}{|c|c|} \hline \text{男生总数} & \text{通过的男生} \\ \hline 3 & 2 \\ \hline 6 & 4 \\ \hline 9 & 6 \\ \hline \end{array} \qquad \begin{array}{|c|c|} \hline \text{女生总数} & \text{通过的女生} \\ \hline 4 & 3 \\ \hline 8 & 6 \\ \hline \end{array} \] 男生通过人数第一次等于女生通过人数是在两者都为6时。学生人数的最小可能值是$9+8=17$。

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